# How does binding energy work?

I do not have a very advanced understanding of binding energy and atomic mass but in my classes, I have learned that the atomic mass is equal to the sum of the number of protons and the number of neutrons. But I know this to not be the case every time because during fusion energy is emitted due to the mass of the product (say, Helium-4) being lighter than the nuclei that you started with (say, 2 protium nuclei). In a book (The Universe in your hand), I read that the difference in mass from which the energy is emitted (due to E=mc^2) is actually from the mass of the mesons (two quarks?) and gluons that carry the strong force between the quarks and between protons and neutrons. According to the book, when fusion occurs the number of gluons and mesons required to hold the atom together is less (the binding energy?), and therefore the total mass decreases.

Now my question is, why does this fusion cause a decrease in mass? For example, why does a hydrogen-2 nucleus have less binding energy, i.e. have fewer mesons and gluons, than two hydrogen-1 nuclei? The hydrogen-1 nuclei should not require any mesons since they only consist of a proton, don't they? If someone could explain this I would be very grateful.

Please inform me of potential errors. :)

So the energy of the atom in the ground state is $$m_pc^2+m_ec^2+BE$$ where BE=-13.6 eV. Note the minus sign.
Now suppose the hydrogen atom has a neutron in addition to the proton. Then we get new terms in the sum of the energy. We have to add $$m_nc^2$$ for the mass energy of the neutron. But then the neutron is bonded to the proton via the strong force, we have a new Binding Energy term. There's also some angular momentum terms now that might not show up in the individual rest masses or the binding energy.